Simulation of Optics Correction for ERLs. V. Sajaev, ANL

Transcription

1 Simulation of Optics Correction for ERLs V. Sajaev, ANL

2 Introduction Minimization of particle losses in ERL arcs requires use of sextupoles As in storage rings, trajectory errors in the presence of sextupoles lead to optics distortion and coupling In synchrotron light sources, the optics is corrected to a very high accuracy using response matrix fit Here we simulated optics correction using response matrix fit for an unclosed beamline to test what kind of correction accuracy can be achieved CEBAF performed optics correction many years ago using same approach but with limited scope and accuracy 1 1 V. Lebedev et al., PAC 97 V. Sajaev, Simulation of Optics Correction... 06/11/2009 2

3 Orbit Response Matrix Fit (ORM) ORM is well known and widely used method for optics correction in storage rings Theoretically, there is no significant difference between ORM for closed and non-closed beamlines: x( s) θ β βθ sin( ψ ψ θ ) = s s θ x( s) = β s βθ cos( ψ s ψ θ πν ) 2sin( πν ) Measured trajectories in both cases depend on beta functions and phase advances and therefore can be used to derive linear optics The main practical difference is that the ORM for a non-closed beamline is triangular with zeros in the top right triangle V. Sajaev, Simulation of Optics Correction... 06/11/2009 3

4 Response matrix fit at APS Response matrix fit is used at APS routinely for lattice correction after fill pattern and lattice changes and also for other measurements like local impedance or local chromaticity The fitting program and GUI are written in Tcl/tk and use elegant 1 for orbit calculations and SDDS toolkit 2 for data processing We added an option for working with non-closed trajectories We used only APS portion of the ERL for simplicity 1 M. Borland, APS-LS 287, M. Borland et al., Proc. of PAC 2003 V. Sajaev, Simulation of Optics Correction... 06/11/2009 4

5 Choice of correctors for measurements For circular machines: Presently, APS has 320 correctors in each plane. Only small fraction of them are used for measurements 27 correctors in each plane. The main requirement is to spread correctors over some phase advance while avoid placing correctors at nπ phase advance For linear machines: Measured trajectory is only affected by elements that are located after the steering magnet, therefore different trajectories provide different amount of useful data One would want to use as many correctors in the beginning of the beamline as possible while keeping them at some phase-space distance V. Sajaev, Simulation of Optics Correction... 06/11/2009 5

6 Choice of correctors Most natural choice for a beamline only two correctors in the beginning of the beamline that are used to scan the phase space. Generate a number of equally-spaced points on the phase-space ellipse at the beginning of the beamline and calculate respective angles and positions at the exit of the second corrector Calculate corrector strengths that would generate the same angles and positions and use these strengths for measurements This way one can generate as many different trajectories as required while still using only correctors in the beginning of the beamline Turns out 1 that was the original idea for the response matrix fit method 1 J. Corbett, private communications V. Sajaev, Simulation of Optics Correction... 06/11/2009 6

7 Simulation procedure Elegant parameter file is generated with element errors The trajectory in the beamline is corrected Measured response matrix and dispersion are calculated on the corrected orbit Response matrix fit is calculated (dispersion included); 30 trajectories were used Quadrupole and skew quadrupole corrections are applied and the resulting beta functions are compared with the ideal beta functions The entire process was run 100 times V. Sajaev, Simulation of Optics Correction... 06/11/2009 7

8 Errors used in calculations Quadrupole gradient error 0.1% Quadrupole tilt error rad Sextupole X and Y displacement 0.5 mm Corrector calibration error 2% Corrector tilt rad BPM calibration error 5% BPM tilt rad BPM measurement noise 1 µm Errors are Gaussian-distributed with two-sigma cut off Sextupole displacements are chosen large based on APS experience and policy of allowing local steering for user beamlines V. Sajaev, Simulation of Optics Correction... 06/11/2009 8

9 Typical beta functions and dispersion before correction Beta X Beta Y X dispersion Y dispersion V. Sajaev, Simulation of Optics Correction... 06/11/2009 9

10 Optics errors before correction For every seed, the beta functions (and dispersion) in the middle of ID straight sections are compared to the ideal lattice and rms difference is calculated. This represents one value for the histogram calculation V. Sajaev, Simulation of Optics Correction... 06/11/

11 Optics correction At APS all quadrupoles have separate power supplies. Therefore, the straightforward way is to apply gradient corrections opposite to those obtained in the ORM fit In real life we use another approach we correct beta functions obtained in the ORM fit using beta function response matrix. This way we can control the strength of the quadrupole corrections that are applied and the accuracy of the correction In these simulations we used this straightforward approach to simplify simulation process V. Sajaev, Simulation of Optics Correction... 06/11/

12 Optics correction results We used several ORM fit configurations First and most straightforward all quadrupoles are used in the fit (Case 1 in the table below) The results were not good; after analyzing differences between seed errors and fit results we noticed that the biggest discrepancies were in the beginning of the lattices To limit quadrupole changes in the beginning of the lattice, we included quadrupole strength constraints in the fit and put heavier constraints on the quadrupoles in the first sector. The improvement was significant Errors Variables Quadrupole constraints β x rms (%) β y rms (%) Case 1 All All No Case 2 All All Yes Case 3 All No half S1 No Case 4 No S1 No S1 No V. Sajaev, Simulation of Optics Correction... 06/11/

13 Optics correction results Why the correction is not perfect? Turns out that the errors in the beginning of lattice cannot be determined precisely enough But it can be compensated by changing incoming beam parameters (if one can measure beta functions downstream Adjustment of incoming beta functions V. Sajaev, Simulation of Optics Correction... 06/11/

14 Piece by piece During our simulations, we have also found that if the focusing errors of the lattice are large enough, sometimes the response matrix fit does not converge We have tested the following procedure that helps in case of convergence problem: split lattice in pieces and perform response matrix fit piece by piece and apply corrections from piece-by-piece solutions This piece by piece approach will probably have to be used at larger facilities to avoid long measurements and huge matrices V. Sajaev, Simulation of Optics Correction... 06/11/2009

15 Coupling correction APS has only 19 dedicated skew quadrupole correctors; this number is adequate for coupling correction to a level of 1% for storage ring beam This turns out to be insufficient for coupling correction of the small ERL beam spurious vertical dispersion could not be corrected well enough Increasing number of skew quads to 80 solves the problem V. Sajaev, Simulation of Optics Correction... 06/11/2009

16 Conclusions We have simulated optics correction for non-closed beamlines using response matrix fit We used 2-corrector measurements to maximize the amount of data obtained in RM measurements We have found that response matrix fit can be used to measure and correct linear lattice successfully We have tested the piece-by-piece approach when one can measure and correct only a part of non-closed beamline which might be useful for large ERLs Since the errors in the beginning of the beamline are harder to determine, one might use piece-by-piece approach with overlapping pieces V. Sajaev, Simulation of Optics Correction... 06/11/